On the Schr\"odingerization method for linear non-unitary dynamics with optimal dependence on matrix queries

The Schr\"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr\"odinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schr\"odinger-type equation in one higher dimension \cite{Schrshort,JLY22SchrLong}. We show that by employing a smooth initialization of the warped phase transform \cite{JLM24SchrBackward}, Schr\"odingerization can in fact achieve optimal scaling in matrix queries. This paper presents the detailed implementation of three smooth initializations for the Schr\"odingerization method: (a) the cut-off function, (b) the higher-order polynomial interpolation, and (c) the Fourier transform methods, that achieve optimality for (a) and near-optimality for (b) and (c). A detailed analysis of key parameters affecting time complexity is conducted.